Reduction of servo following errors in position and velocity control systems of the iteratively computing type

ABSTRACT

Methods and apparatus for reducing following errors in the servo drives of positioning and velocity control systems wherein the commanded position is signaled and updated by increments Delta X during each successive equal time periods Delta T, -so as to cause the controlled member movable along any axis X to travel at a velocity Vx Delta X/ Delta T. The commanded position is updated by iterative computations at least once during each period Delta T, and the Delta X increments to be used for such updating are numerically signaled and can be changed as frequently as once during each period Delta T. In order to create a feed forward signal for application to the servo in a sense additive to the position command signal, the digitally signaled number Delta X is converted into a corresponding analog signal. In the preferred arrangement, this conversion is accomplished by a counter which is preset to the number Delta X at regularly recurring instants and arranged to count down whenever the number held in the counter is other than zero. A decoder responsive to a predetermined count, for example zero, not only terminates the counting action until the next presetting but also produces a constant frequency, variable duty cycle squarewave having an average or dc. value proportional to the number Delta X and which serves as an accurate feed forward input to the servo.

United States Patent [191 Simon et al.

[ Mar. 19, 1974 REDUCTION OF SERVO FOLLOWING ERRORS IN POSITION ANDVELOCITY CONTROL SYSTEMS OF THE ITERATIVELY COMPUTING TYPE [75]Inventors: James B. Simon; Thomas B. Bullock,

both of Fond du Lac, Wis.

[73] Assignee: Giddings 8L Lewis, Inc., Fond du Lac, Wis.

22 Filed: Feb. 27, 1973 21 Appl. No.: 336,268

[52] US. Cl.. 235/151.11, 235/150.1, 318/573 [51] Int. Cl G06f 15/46,GOSb 21/00 [58] Field of Search 235/150.1, 151.11

[56] References Cited UNITED STATES PATENTS 2,829,329 4/1958 Silva235/151 Primary Examiner-Eugene G. Botz Attorney, Agent, or FirmWolfe,Hubbard, Leydig, Voit & Osann, Ltd.

[ ABSTRACT Methods and apparatus for reducing following errors in theservo drives of positioning and velocity control systems wherein thecommanded position is signaled and updated by increments AX during eachsuccessive equal time periods AT, so as to cause the controlled membermovable along any axis X to travel at a velocity V AX/AT. The commandedposition is updated by iterative computations at least once during eachperiod AT, and the AX incrementsto be used for such updating arenumerically signaled and can be changed as frequently as once duringeach period AT. In order to create a feed forward signal for applicationto the servo in a sense additive to the position command signal, thedigitally signaled number AX is converted into a corresponding analogsignal. In the preferred arrangement, this conversion is accomplished bya counter which is preset to the number AX at regularly recurringinstants and arranged to count down whenever the number held in thecounter is other than zero. A decoder responsive to a predeterminedcount, for example zero, not only terminates the counting action untilthe next presetting but also produces a constant frequency, variableduty cycle squarewave having an average or dc. value proportional to thenumber AX and which serves as an accurate feed forward input to theservo.

14 Claims, 8 Drawing Figures urmmze 4/0.

15m flame:

PATENIED MAR 19 I974 SHEEI10F4 PATENFHI m I 9 m4 SHEET 3 0f 4hilllllll'lllllll REDUCTION OF SERVO FOLLOWING ERRORS IN POSITION ANDVELOCITY CONTROL SYSTEMS OF THE ITERATIVELY COMPUTING TYPE BACKGROUND OFTHE INVENTION 1. Field of the Invention The present invention relates ingeneral to closed loop servo systems for driving a movable member alonga path (i.e., along an axis) to keep its position dynamically inagreement with a changing position command and to make its velocitysubstantially equal to the rate of change of the position command. Moreparticularly, the invention relates to such servo systems used with orforming a part of numerical control systems of the type which generate adigitally represented numerical position command changed by iterativecomputations repeated during each of successive equal time periods AT,and the invention deals with the production and utilization of feedforward signals applied as supplemental inputsto the servos to reduce oreliminate following errors.

2. Description of the Prior Art The basic concept of feed forward toreduce following errors in positioning servo systems is per se wellknown. Although no attempt has been made to collect and list prior artarticles or publications in the technical literature pertaining to feedforward in servo controls, the reader may gain a general familiarizationwith the prior art by referring to Document AD 688 798 distributed byClearinghouse for Federal and Scientific Information, Springfield, Va.2215] (such document being Technical Report No. 8 by John A. Miller etal. under Air Force Contract F-44620-68-C-0O2l) and to the bibliographycontained in such document.

Moreover, it has heretofore been proposed to create a supplemental feedforward signal proportional to the desired velocity, and to inject thatsignal into a servo loop forming a part of a numerical control system,as disclosed in U.S. Pat. No. 3,539,897 issued in the name of M. R.Sommeria. According to the disclosure of the latter patent, however,such a feed forward signal is employed in a numerical control system ofthe command pulse type wherein command pulses of equal weight aregenerated with varying or changeable frequencies in order to cause amovable element to travel through desired distances and at desiredvelocities along one or more axes.

SUMMARY OF THE INVENTION It is the general aim of the invention toreduce the following error in positioning servo systems of the typewherein a changing position command signal is periodically incrementedby changeable amounts during each of successive, equal time periods andto do so by creating a feed forward signal injected into the servo loopin a very simple, expeditious and low cost fashion.

A more specific object of the invention is to make advantageous use ofchangeable, digitally signaled numbers, and timing generator signals,which already exist in a numerical control director of the iterativelycomputing type disclosed in U.S. Pat. No. 3,656,124, in order to producefeed forward signals which are proportional to the individual axisvelocities at which a controlled member is to be moved so as to make itsresultant velocity equal to that designated by numerical programinformation.

Still another object is to reduce servo following errors along any axisX in a numerically controlled positioning system by creating a feedforward signal which is an analog counterpart of a digitally signaledincremental number AX, such analog signal accurately tracking changes inthat number, and such number being that which is added or subtracted toa position command number during each of successive, equal time periodsAT in order to cause a controlled member to move progressively along theaxis at a desired velocity.

These and other objects and advantages will become apparent from thefollowing description, taken with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS The drawing FIGS. 1 through in U.S.Pat. No. 3,656,124 (herein called Case A) are hereby incorpo rated byreference, and a brief description of such figures at Columns 35 of thatpatent are similarly incorporated. In order that reference may be madeherein to both the drawing figures of Case A and those drawing figureswhich are supplementally included with the present application, thosefigures which are submitted with the present application will bedesignated by alphabetical characters, namely:

FIG. A is a generalized block diagram of a numerical control system ofthe type disclosed in detail by Case A.

FIG. B is a very diagrammatic but conventional representation of aclosed loop positioning system;

FIG. C is similar to FIG. B but shows such a positioning system with twoinputs for the purpose of analyzing the effect of a supplemental feedforward" input signal;

FIG. D corresponds to a portion of FIG. C and indicates the manner inwhich a supplemental time derivative signal may be applied to a servoloop to create the desired feed forward action;

FIG. E is a block diagram in general form illustrating an embodiment ofthe present invention;

FIG. F is a graph of voltage versus time showing the manner in which thedc. value of a squarewave voltage, having constant periodicity but avariable duty cycle, changes; and

FIGS. Ga and Gb, when joined as indicated, form a composite FIG. G whichis a block diagram, partly in schematic circuit form, showing the systemof FIG. E in greater detail.

While the invention has been shown and will be described with referenceto a specific and preferred embodiment of the apparatus and methodsteps, there is no intention that the invention thus be limited to suchdetail. On the contrary, it is intended here to cover all alternatives,modifications and embodiments which fall within the spirit and scope ofthe appended claims.

DESCRIPTION OF THE PREFERRED EMBODIMENT CROSS REFERENCE TO U.S. PAT. NO.3,656,124

In order to make the present specification relatively brief and avoidredescription of both the environment of the present invention and theknown elements of an iteratively computing numerical control systemwhich forms a part of the present invention, the specification atColumns 1 through 144 of U.S. Pat. No. 3,656,124

(herein called Case A) is hereby incorporated by reference and made apart hereof. In the description which follows, it will be assumed thatthe reader is familiar with the drawings, terms and symbols of Case A.Primary reference will be made to the system illustrated in FIGS. 400through 40m in Case A, but these latter figures are best understood byfirst becoming familiar with FIGS. 9a through 91.

BRIEF REVIEW OF THE ITERATIVELY COMPUTING NUMERICAL CONTROL SYSTEM ANDITS SERVOS FIG. A is a general representation of the iterativelycomputing numerical control system which is described in more detail byCase A. By way of review, a clocked, iteratively computing director 1001receives successive sets or blocks of programmed numerical signals froma tape reader 41 (FIG. 40m) which senses a record medium such as apunched tape 37 (FIGS. 3 and 4). Each block of such data defines alinear or circular path segment (see FIG. 2) to be followed by a movablemember or cutter 14 in traveling relative to a workpiece 13 (FIG. 1).The composite or path segment motion is the resultant of simultaneousvector component movements along one or more of three axes X, Y and Z.The individual axis components of the cutters displacement pursuant toany segment are executed by dynamically changing the individualpositions of elements such as the saddle 16, the ram 15 and the table 11ofa machine tool (FIG. 1) driven by individual axis servo systems whichinclude motors 21, 25 and 23 coupled by means such as lead screws 20,24, 22 to those elements. By coordination of the individual axisvelocities, the composite or resultant path segment velocity is made toagree with a desired, programmed path feed rate.

It will be recalled that the director 1001 includes a time basegenerator (FIG. 912 or FIG. 40b) which measures off successive, equaltime periods AT. In the specific example given, these periods are each1/50 or 0.020 seconds. Each such period AT spans the interval requiredfor the three decade counters 133, 135, 137 in tandem (FIG. 40b) tocount from the decimal state 000 to the state 999. Separate time stepsof microseconds duration are individually measured off within eachperiod AT, and each time step is separately identified by the signalingof a number 000 to 999 on the decoder output terminals marked in FIG.40b.

During any of the successive periods AT, the director may operate inMode 1, 2, 3 or 4 but for purposes of the present invention, this is ofno consequence and only the common or all modes sequential operations asset out in Table VII at Column 111 in Case A need be considered. Thedirector may also operate in either a linear or circular interpolationmode, but the operation of the present invention will be equallyeffective and advantageous in either of such modes.

Without here repeating the details of Case A, it may be noted that thedirector shown in FIG. A produces sets of digital signals which in BCDnotation represent the numerical values of changeable servo commandnumbers XSC, YSC (and ZSC if a third axis is used). See the registers121' and 119' in FIG. 40g. These registers have the numbers XSC and YSCincremented (increased or decreased) by amounts AX and AY (calledmacromoves) during each period AT by the repeated operation of the timeshared digital computer 53' (FIG. 40]). The computer input and outputtrunks CIT and COT extend to WRITE and READ gates associated with thevarious storage registers included in the system, and each such gate iscontrolled by a pulse gate array (PGA) so that it is opened on certainpredetermined time steps within successive AT's and according to themode in which the system is then operating. It will therefore berecalled that in the preferred operation described in Case A, the numberheld in the XSC register 121' is incrementally changed by an amount0.1AX (called a micromove) at ten equally spaced instants within eachperiod AT, as is made clear by Table VII, but the result is to make thenumber XSC signaled at the output terminals of the register 121' changeby an incremental amount A X during each period AT, i.e., such that XSC,XSC, AX. The number YSC is similarly incremented by the amount AY duringeach period AT.

The macromove numbers AX and AY (and also AZ if a third axis isemployed) are normally computed from time-to-time and indeed as often asonce during each AT. Thus they are changeable numbers. Their valuesdepend upon both (a) the path velocity at which the main controlledmember or cutter 14 is to move relative to the workpiece 13, and (b) theangle 6 which the path segment makes with the X axis. These numbers AXand AY are numerically represented in BCD notation by respective sets ofdigital output signals from storage registers 109' and 107' in FIG.4011. The computer 53 coupled via PGAs and READ and WRITE gates to theseAX and AY registers causes the macromove numbers AX and AY to be changedperiodically under certain conditions as shown for example in TableVIII.

As represented in FIG. A, the changeable position commands digitallysignaled as numbers XSC, YSC, ZSC are supplied as inputs todigital-to-analog converters 1002, 1003, 1004. The outputs of the latterare analog signals XC, YC, ZC which vary according to changes in thedigital position command numbers XSC, YSC and ZSC. In the preferredembodiment of Case A, each of these digital-to-analog converters isconstituted by a compare circuit (see 533 in FIG. 40d) which receivesthe signaled position command number and also receives digital sweepnumbers from FIG. 40b, so as to act as a digital-to-phase converter. Inother words, each of the compare circuits produces a pulse during eachyc f a d ital w p h phhas a frcqqensm i Hz., with the time location orphase of that pulse within each cycle spaced from the cycle mid-pointaccording to the then-existing value of the digital input number XSC orYSC. The phase of the compare output pulses thus represents in analogform the desired position of the controlled member along thecorresponding axis. But further, the timing signal generator of FIG. 40balso produces from a wave shaper 181 a sinusoidal reference voltagehaving a frequency of 500 Hz., and in phase with the digital sweep, asshown in FIG. 10b. This reference voltage is applied as an excitationsignal to feedback transducers associated with the element movable alongeach axis, such transducers being here shown in FIG. A as resolvers R,,,R,,, R The rotors of these resolvers are directly or indirectly coupledto the output shafts of the corresponding axis servo motors 21, 25, 23and their output windings thus produce sinusoidal feedback signals XF,YF, ZF which, in effect, represent the actual axis positions by theirphase relative to the reference wave. The analog position command signalXC and the feedback signal XF are supplied as inputs to a servo systemgenerally represented at 1005 in FIG. A. The position command signalsand F= A/s (C-F) feedback signals are similarly applied to servoamplifi- 5 And by re-arrangement, Equation (3) becomes:

ers 1006 and 1007 for the Y and Z axes. In each case, the servoamplifier acts to subtract the actual position signal XF from theposition command signal XC and to amplify the difference or error. Theamplifier output M, is applied to the motor; it is a voltage which isproportional to the difference or error E XC XF. Thus, themotor-energizing voltages labeled M M,, and M in FIG. A areinstantaneously proportional to the position errors, and they thus causethe driven elements l6, l5 and 1 l to move at velocities which areproportional to the respective axis position errors.

As will be apparent, in order to make the controlled elements move atrelatively high velocities, the position errors must be relativelylarge. This position error is called the following error and it may bedifferent for different axes because the individual axis velocities maybe widely different at any one time, depending upon the angle of thecomposite path segment. In consequence, the accuracy with which thefinal controlled member (the cutter 14) tracks the desired path segmentmay be adversely degraded to a significant degree.

ELEMENTARY REVIEW OF FEED FORWARD ACTION Referring to the very simplerepresentation of a closed loop, negative feedback positioning servosystem in FIG. B, it may be assumed that an input command signal C isapplied through an algebraic summing device 1010 to a combined amplifierand actuator 101 1 which may include a motor and suitable elements suchas gears coupling it to the driven element. The output of theamplifier-actuator is here labeled F and represents the actual positionof the controlled element. By appropriate feedback means, the output Fis applied subtractively to the summing device 1010 so that the input tothe amplifier-actuator may be deemed as equaling the difference error Ebetween the input command C and the output F. In a very simplifiedsense, the transfer function of the amplifier-actuator 101 1 may beconsidered as equal to A/s where A represents the gain and s is thedifferential operator (representing d/dt) employed in LaPlacetransforms.

The applicable equations for FIG. B may be written to illustrate theexistence of a following error in a conventional positioning servosystem. To begin, it is apparent that the output F of theamplifier-actuator 1011 is equal to its input multiplied by its transferfunction:

But the error is simply equal to the difference between the input C andthe feedback signal F, i.e.:

E C F; C E F By substituting Equation (1) into Equation (2), oneobtains:

F(l +A/s)= AC/s FS'i'FA=AC If the value of C from Equation (2) issubstituted into Equation (30), the result becomes:

Fs+FA=A(E+F) And by re-arrangement and solution for E, this reduces to:

It is apparent, therefore, that in a conventional positioning servosystem the error E is finite and proportional to the rate of change ofthe output F. Even though the error E can be held relatively low bymaking the gain A large, there are limits imposed upon the magnitude ofgain which can be used because of stability considerations.

If, however, a supplemental input signal is injected into the servosystem of FIG. B-as here represented in FIG. C-then the following errorcan be reduced and made to approach practically zero. Assume for themoment that the same command signal C is passed through an amplifier orsignal transforming circuit 1012 having a transfer function which is theinverse of that for the amplifier-actuator 1011, namely, s/A. Assumefurther that the output from the unit 1012, which is sC/A, is applied toa second summing device 1013 so that it is combined additively with theerror signal E discussed above with reference to FIG. B. In thisarrangement, the applicable equations may be written by recognizing thatthe output F is equal to the input applied to the amplifier-actuator1011 multiplied by the latters transfer function, and that input is thesum of the two signals applied to the summing device 1013. Thus, one maywrite:

F= EA/s sC/A- A/s EA/s C The error signal E is related to the inputsignal C and the feedback signal F as above stated, namely:

E C F; F C E (2') By substituting F from Equation (1') into Equation(2'), one obtains:

C-E=EA/s+C which simplifies to:

EA/S+E=O E(A/s+l)= Equation (5b) on its face seems to indicate a secondsolution by which the quantity A/s is equal to a l but that solution isobviously invalid because the gain A is always a constant having apositive finite value. Thus, from Equation (50) it is clear that theinjection of a signal sC/A additively combined with the error E of thepositioning servo can result in the following error being heldsubstantially at zero as the input and output signals have differentrates of change representing different commanded and actual velocities.

FIG. C makes it plain that the auxiliary signal to be injected into theservo can take the form sC applied to an amplifier or attentatingelement 1012 having a gain of HA, and this will result in the creationof a signal sC/A for injection into the summing device 1013. Recallingthat s represents a differential operator, the signal sC as shown inFIG. D thus may be one which varies as the rate of change of the mainposition command signal C, i.e., the signal sC may be one whichrepresents a desired velocity corresponding to the rate of change of thedesired position command signal C.

A PREFERRED EMBODIMENT OF THE PRESENT INVENTION In accordance with thepresent invention, it has been discovered that in an iterativelycomputing numerical control system of the type here described, thedigitally signaled increment number for any axis (that is, AX, AY orAZ), continuously available but changeable in value, is always directlyproportional to the velocity at which the element is to be driven alongthat axis; and provision is made to derive from that signaled incrementnumber a proportional signal which is injected into the axis servo loopto create feed forward action and reduction or elimination of thefollowing error.

More particularly, and in the preferred embodiment shown in FIG. E forthe X axis (substantial duplicates, not shown, being employed for anyother axes such as Y and Z), the number AX signaled by the numericalcontrol director 1001 is converted into a correspondingly varying analogdc. voltage. This voltage is added to the error signal E in the X axisservo loop. Thus, by the addition of only a few relatively simpleapparatus components to the existing structure of the iterativelycomputing director and the servo system, the advantages of feed forwardcompensation are achieved.

Specifically in carry out the invention, a presettable digital counter1015 is provided together with means responsive to the digital signalsrepresenting AX to preset that counter to a count state or number Nwhich differs from a predetermined count state or number P by themagnitude of the number AX. This presetting is performed once duringeach of successive equal time periods AT of predetermined duration butwhich in any event are shorter than the iteration periods AT. In thepreferred arrangement shown, the count state N is equal to AX and thepredetermined count state P is zero, so by directly presetting thenumber AX into the counter, the difference is NP= AX O AX. Asillustrated in FIG. E, the presettable counter 1015 has presetting inputterminals connected to a multi-conductor input trunk 1015a forming theoutput of presetting gates 1016 which receive as their inputs theconductors of the computer output trunk COT existing in the director1001. The presetting gates are normally closed, but they are enabled oropened once during each 1/500 second by a gate pulse source 1017, thus,making the shorter period AT in this example equal to 1/500 second. Aswill be explained hereinafter, the gate pulse source shown in FIG. E maybe easily constructed as a pulse gate array of the type employed in thedirector 1001, and it can readily be controlled to open the gates 1016five hundred times during each second.

Further in keeping with the invention, the counter 1015 is associatedwith means to make it always receive and count recurring pulses of apredetermined frequency whenever it is not in the predetermined countstate P, so that it counts from the state N to which it is preset towardthe count state P. Thereafter the counting operation ceases until thenext presetting operation occurs. As shown in FIG. E, the counter 1015is a downwardly counting counter having output terminals connectedthrough a multi-conductor cable to an associated fir?) decoder 1018. Theorganization of the decoder will be readily known to and understood bythose skilled in the art; it functions simply to produce a binary I"level voltage on its output conductor 1019 whenever the counter is inany count state other than P (e.g., other than zero), This decoderoutput is connected as one input to an AND gate 1020 which receives asits other input a 500 KHZ. train of regularly recurring pulses. In thepreferred arrangement, this train of pulses appears in the timing signalgenerator of the director 1001, and thus need only be supplied by aconnecting conductor to the gate 1020. The output of that gate leads tothe pulse input of the counter 1015. It will be apparent thatimmediately after the number AX haspgen preset into the dow r 1 counter1015, th e latter counts downwardly from count state N (in this example,AX) toward count state P (in this example, zero). When it reaches thelatter count state, the input of counting pulses is terminated byclosure of the N gate 1020. Thereafter, when the gate pulse source 1017again opens the presetting gates 1016 and the number AX is transferredfrom the output trunk COT to again preset the counter to the number AX,the down counting operation is started again. Because the periodicity ofthe recurring pulses is less than the periodicity with which the counter1015 is preset (that is, the pulse frequency is 500 KHZ while thepresetting frequency is 5 1 1 1 9JE E.iEPfiEQLQFPPRidy sash downcounting to zero is accurately proportional to the number AX to whichthe counter is preset before down counting begins.

Further in the practice of the invention, means are provided forproducing a squarewave voltage which has one level when the presettablecounter is in a first predetermined count state (here the count state Pwhich is assumed to be zero) and which has a second level when thecounter is not in that particular count state.

As a surprising and unique advantage of the present invention in theembodiment of FIG. E, this latter means is constituted directly by the2H) decoder 1018, and the bi-level squarewave voltage appears directlyon the output conductor 1019 of that decoder. As illustrated in FIG. E,the conductor 1019 carries a squarewave voltage 1024 having a first orzero volt level L, whenever the counter 1015 contains the predeterminednumber P (in this example, zero); and having a second level L wheneverthe counter holds any other number. Merely as an example, the squarewavevoltage level L, may be zero volts, and the squarewave voltage level L,may be volts. Since the beginning instant t, in each cycle of thissquarewave coincides with the presetting of the counter 1015 to a countstate equalling the number AX, and since the squarewave switches fromthe level L: to the level L, at an instant t spaced by a time interval awhich must be proportional to the number AX, the squarewave 1024 has aconstant frequency (i.e., a period of H500 second), but the dutycycle-that is, the ratio of its high to low level intervals here labeleda and b--varies directly according to changes in the value of the numberAX.

The variable duty cycle squarewave voltage 1024 is passed through anaveraging filter 1025 to produce a dc. signal which varies in magnitudesmoothly as the duty cycle changes. As illustrated in FIG. F, asquarewave voltage which resides alternately at two levels L and L,,always with a constant period T, but with changes in the duration a ofthe level L intervals, has a dc. or averagevalue representableas dottedlines DC, and DC,. When the duty cycle or ratio all; results in a dc.voltage DC and if the ratio a/b increases, then the dc. voltagecorrespondingly receives to a higher value DC The dc. output of thefilter 1025 is applied across a potentiometer 1026 having an adjustablewiper 1026a which is connected to one of the inputs of an algebraicsumming device 1028. The second input to this latter device is the errorsignal E, which is created by another algebraic summing device 1029having as its two inputs (a) the analog X axis position command signalXC and (b) the analog actual position feedback signal XF, the latterbeing created by an appropriate feedback transducer here shown as aresolver R, mechanically coupled to the output shaft of anamplifier-actuator 1030. This amplifier-actuator as shown in FIG. Eincludes a separate amplifier and the X axis servo motor, and may becharacterized as having a transfer function A/s. It receives as itsinput signal the output of the summing device 1028, and its output F maybe designated as the X axis position in which a controlled element, suchas the saddle 16, resides.

By way of a summary description of operation, it may be noted that thenumber AX is continuously signaled in the numerical control director1001, and in particular it is signaled by the output terminals of theregister 109 in FIG. 9h. As indicated by the programming sequences shownin Table VII, the number AX is read from the AX register 109 into thecleared computer 53 10 times during each period AT, i.e., at time step016 in each of Time Columns X00 through X300 and X500 through X900.During Time Column X400 the number AX is also read into the clearedcomputer, but on time step X403. Thus, it is a simple matter to utilizethe next-succeeding time step in each of the Table VII time columns towrite the number AX from the computer output trunk COT into the counter1015 before continuing the subsequent operations which are set out inTable VII. The gate pulse source 1017 supplies an enabling pulse to thepresetting gates 1016 on time step X404 and all of the time steps XX17(except X417) during each AT iteration period represented by Table VII.In this fashion, the gates 1016 preset into the counter 1015 from theoutput trunk COT the number AX 500 times each second.

Immediately after such presetting of the down counter 1015, thesquarewave voltage on line 1019 rises to a high level L and enables thegate 1020 so that pulses at a frequency of 500 KHz. are transmitted tothe counting input of the counter. The counter than proceeds to countdown and the signal on line 1019 remains at the high level L, until azero count is reached, with the result that the time interval a duringwhich down counting takes place, and during which the squarewavevoltages remain at the level L is proportional to the last-signaledvalue of the number AX. When the squarewave voltage 1024 falls to itslower level L (because the counter 1015 has reached zero count), thegate 1020 closes and counting ceases until the gate pulse source 1017again opens the presetting gates 1016.

The squarewave voltage 1024 is one which varies in its duty cycleaccording to changes in the number AX. The dc. voltage produced by theaveraging filter 1025, therefore, varies in magnitude with changes inthe number AX, and a predetermined fraction of that voltage is tappedfrom the potentiometer 1026 and fed via the wiper 1026A to one input ofthe summing device 1028.

This input signal to the summing device may be designated as a feedforward voltage FFV. It is, as noted, proportional to the signalednumber AX, i.e., FFV=k AX. It will be recalled, however, that the numberXSC signaled in the director 101 is dynamically changed to represent achanging commanded position for the element 16. Indeed. it isdynamically changed by algebraically adding the number AX thereto onceduring each iteration period AT. Thus. the changes in the command signalXSC and its analog counterpart XC may be represented as AC/ AT where ACis equal in value to the number AX. It becomes apparent, therefore, thatAX is directly proportional to the desired velocity at which the element16 is to move along the X axis, but this number is proportional to thefirst time derivative dC/dT. If one uses the derivative operator s, itmay be said that the feed forward voltage FFV is equal to sC/ A, where Ais a factor of proportionality representing gain or attenuation, andwhich may be adjusted in its value by changing the setting of the wiper1026a. Given an existing servo system having an amplifier-motor actuatorwith a transfer function of A/s, it is thus a simple matter to adjustwiper 1026a so that the feed forward voltage FFV is equal to sC/A or, inother words, proportional to the desired velocity which is representedby the then-existing rate of change of the input position command signalXSC.

FIG. G shows in greater detail the apparatus generally depicted in FIG.E, and illustrates how the feed forward voltage FFV may be created witha plus or minus polarity whenever the AX number is positive or negative(and the XSC number is being increased or decreased during each periodAT). As will be apparent,

the down counter 1015 is organized specifically to include four tandemcounter decades A, B, C, D each having four presetting input lines andfour output lines, each signaling in BCD notation any decimal digitvalue between and 9. The decimal point is implicitly located one placeto the left of decade A, so that the counter can be preset to hold anyAX number from 0 to 0.09999. The presetting WRITE gates 1016 arearranged in four four-part sets each corresponding to one decade or onedecimal digit place, each individual input terminal of these presettinggates being connected to one of the COT conductors and the individualoutput terminals of these gates being connected to the correspondingpre-setting inputs of the counter. The gates are all enabled or openedby an enabling signal from a PGA 1017 which has its inputs connected totheterminals of FIG. 40b such that it produces an enabling signal ontime steps X417 and X404 of every period AT. The organization of thediodes in the PGA 1017 will be readily apparent to those skilled in theart from what has been said above. For the sake of completeness it maybe considered that the legend SAR presently appearing in Table VII fortime steps X017, X117, X217, X317, X517, X617, X717, X817 and X917- ismodified to read W/PSC,SAR, where PSC stands for presettable counter."Thus, on the seventeenth step within every time column of a AT period(excepting Column X400) the number AX which has previously been readinto the cleared computer is written into the counter 1015. Similarly,the legend which appears at time step X404 in Table VII is modifled fromits previous form R/XCP to read W/PSC, R/XCP. This assures that thecounter 1015 is preset to the number AX during every time column X400 ofeach measured period AT, and it is thus so preset times per period AT orat a rate of once each l/SOO seconds.

The AND gate 1020 shown in FIG. G receives one input from the output ofthe zero decoder 1018 via line 1019 as previously described. However, inFIG. G it is indicated that the source of recurring pulses at a 500 KHz.frequency supplied to the AND gate 1020 may be readily taken from thetiming signal generator which is illustrated in FIG. 40b. Specifically,the output of the divide-by-four circuit 141' in FIG. 40b is a train ofpulses having a frequency of 500 KHZ. Thus, in addition to being passedinto the A decade counter 131' in FIG. 40b, this train of pulses is alsocoupled by a single conductor to the second input of the AND gate 1020.Thus, whenever the gate is enabled, it passes relatively high frequency(500 KHZ) pulses to the counting input of the down counter 1015.

That portion of FIG. G which is enclosed within dashed lines correspondsessentially tothe apparatus shown in FIG. 40a, and it has been solabeled in FIG. G. The same reference characters which appear in FIG.400 are here employed, so that the operation of this portion of theapparatus will already be understood. Briefly stated, however, thecompare circuit which is shown in FIG. 40d produces a variable phasecompare pulse (as noted above) in response to input signals XSC whichare supplied from the register 121 which is shown in FIG. 40g. Thisvariable phase compare pulse is labeled XC in FIG. G and is an analogphase variation which from instant to instant designates the commandedposition to which the controlled element 16 is to be moved. Thesecompare pulses are passed to the set input of a flip-flop 535' which, asdescribed above, functions as a phase discriminator, comparing phase ofthe pulses XC with the feedback signals produced by the resolver 523 asprocessed through a 5 phase shifter 531 and a square wave shaper 537That is, the flip-flop 535' produces at its 1 and 0" output terminalscomplemental squarewaves having duty cycles and average dc. values whichvary according to the position error, i.e., the difference between thephase of 0 the pulses XC and the phase of the feedback wave produced bythe resolver 523 and phase shifter 531'. As confirmed by FIG. G, theresolver 523 has its rotor coupled to the output shaft of the servomotor 21, so that a phase variable analog signal representing the actualposition of the movable element 16 is fed back to the phasediscriminator. It will be recognized that the phase discriminator, whichis formed by the phase shifter 531', shaper 537 and flip-flop 535',functions as an algebraic summing device, that is, the output signalproduced by the selectively enabled current generators 551' and 549'across the resistor 553 is a squarewave having a variable duty cyclewhich changes according to the difference between the input signal XCand the feedback signal F. The filter S61 converts this variable dutycycle squarewave, which can have either positive or negative pulseportions, into a relatively smooth dc. voltage proportional to the errorE, i.e.,

proportional to the difference between the command signal XC and thefeedback signal F.

To complete the feed forward portion of the apparatus shown in FIG. G,and to permit the feed forward voltage FFV to be positive or negativewhen the number AX is positive or negative, a switched flip-flop V5 iscontrolled by signals representing the sign of the number AX and set tothe proper one of its two possible states by the output pulse from thePGA 1017 every 1/500 second. It will be recalled that the COT()conductor which is included within the computer output trunk COT residesat a binary 1 level when the number contained in the computeraccumulator has a negative sign, and it resides at a binary 0 level whenthat number has a positive sign. Thus, at those instants (describedabove) when the number AX is in the accumulator, the conductor COT()will reside at a binary 0 or 1 level if the sign of the AX number ispositive or negative. This conductor COT() is coupled directly throughthe reset control terminal R of the switched flip-flop VS, and it iscoupled to an inverter 1030 to 50 the set control terminal S. Inaddition, the output of the PGA 1017 is connected to the switch inputterminal SW of the flip-flop so that at those instants when the PGAproduces an enabling pulse, the flipflop VS is switched to its binary 1or its binary 0 state if the number AX is positive or negative in sign.The signals VC+ and VC- appearing on conductors connected to the Q and Qoutputs of the flip-flop VS thus respectively reside at binary 1 levelsif the sign of the AX number last preset into the counter 1015 ispositive or negative. These outputs are connected respectively ascountrolling inputs to two AND gates 1031 and 1032 which both receive astheir second inputs the squarewave 1024 produced by the decoder 1018.Therefore, during those intervals when the squarewave 1024 is at itshigh or binary 1 level L the gate 1031 will produce an output responseif the AX number has a positive sign; but conversely during those timeintervals when the squarewave 1024 is at its high level L the gate 1032will produce an output signal if the AX number has a negative sign.

The outputs of the gates 1031 and 1032 are respectively supplied ascontrolling signals to switchable, constant current generators 1033 and1034, respectively, to turn the latter on or off. These two constantcurrent generators are supplied respectively from positive and negativevoltage sources, so that when the generator 1033 is turned on a positivevoltage drop will appear across a common series resistor 1035, and whenthe generator 1034 is turned on a negative voltage drop will appearacross that resistor. In either case, the amplitude of the voltageappearing across the resistor 1035 is limited and held constant at apredetermined value established by a bi-directional Zener diode 1036.Thus, the output across the resistor 1035 is a squarewave having avariable duty cycle identical to that of the squarewave having avariable duty cycle identical to that of the square wave 1024, but theon pulses within that squarewave will be of positive polarity if thesign of the AX number is positive and will be of negative polarity ifthe sign of the AX number is negative. In effect, therefore, a bi-polarvariable duty cycle squarewave is created with the on periods determinedby and made proportional to the magnitude of the AX number, and with thepolarity of the squarewave being determined by the sign of the AXnumber. This bi-polar squarewave is passed through the smoothing oraveraging filter 1025 and thence supplied across the potentiometer 1026,so that the final feed forward voltage FFV appearing on thepotentiometer wiper 1026a is proportional to the number AX and agreeablein polarity with the sign of that number. The factor of proportionalitymay, of course, be adjusted by setting the position of the wiper 1026a.

For the purpose of adding the feed forward voltage FFV to the errorsignal or voltage E, these two voltages are applied through inputresistors 1040 and 1041 to the input terminal of a combined algebraicsumming and amplifying device, here shown as a summing operationalamplifier 521 having a negative feedback path formed by a capacitor1042. As is well known, such an operational amplifier 521 functions toproduce an output signal here labeled M, which is eqaul to the weightedsum of the input signals, the weighting being determined by the relativemagnitudes of the resistors 1040 and 1041. To impart stiffness to theservo response characteristics, a tachometer 1044 is coupled to bedriven by the motor 21', and its dc. voltage (proportional to motorspeed) is also applied in an additive sense through a resistor 1045 tothe input terminal of the amplifier 521'.

In summary, the apparatus illustrated by FIG. G employs a successivelypreset down counter 1015 to pro duce a variable duty cycle squarewave1024 which has a dc. or average value proportional to the number AX.This dc. average value, as noted above, is therefore proportional to thedesired velocity at which the controlled element 16' is to be moved, andit is proportional to the rate of change of the position command signalsXSC (in digital form) and XC (in analog form). The squarewave 1024 is,however, converted into a bipolar squarewave by the controlled flip-flopVS, gates 1031 and 1032, and the constant current generators 1033, 1034.The bi-polar variable squarewave signal appearing across the resistor1035 is converted irito a corresponding dc. variation, and thus the feedforward voltage FFV may be of either positive or negative polaritydepending upon whether the position command signal XC is increasing ordecreasing. By proper sizing of the individual circuit components andadjustment of the potentiometer 1026, the feed forward voltage can bemade to represent the quantity sc/A in a system having an amplifier521', motor 21' and drive connections which collectively exhibit atransfer function of A/s. With such adjustments, the system will operatewith substantially zero following error. This improvement is broughtabout by the relatively simple addition of only a few components to aniteratively computing numerical control director.

Those skilled in the art will recognize that an equivalent operation andresult can be obtained by making the counter 1015 an upwardly countingcounter instead of a down counter. If this choice is made, it is onlynecessary that appropriate inverters be interposed just ahead of or justfollowing the gates 1016 so as to preset the counter to the complementof the AX number, and therefore, in effect, to the negative of the AXnumber. In this way, the counter would always be preset to aneffectively negative value and would count up to its zero count statefor an interval of time proportional to the value of the AX number.

we claim:

1. In a positioning system for moving an element to successive positionsalong an axis in accordance with numerical commands, the combinationcomprising a. means for measuring off successive equal time periods AT,

b. means for producing first signals digitally representing a changeablenumber AX, where AX is proportional to the velocity V, at which saidelement is to be moved,

c. means for producing second signals digitally representing achangeable command number XSC and for changing it by the amount AXduring each period AT,

(1. means for converting said second signals into a correspondinglychanging analog command signal,

e. firearms; producing an analog feedback signal representing the actualposition of the element along the axis,

f. means responsive to said first signals for producing an analog feedforward signal proportional to'the changeable number AX,

g. means for combining said feedback signal subtractively and said feedforward signal additively with said command signal to produce a modifiederror signal, and

h. amplifier and motor means responsive tosaid error signal for drivingsaid element along the axis.

2. The combination set forth in claim 1 further including means foradjusting the constant of proportionality relating said feed forwardsignal to the number AX represented by said first signals.

3. The combination set forth in claim 1 further characterized in thatsaid adjusting means is adjusted to make said constant ofproportionality substantially equal to HA, where A is the gain in thegeneralized transfer function A/s of said amplifier and motor means.

4. The combination set forth in claim 1 further characterized in thatsaid means (f) comprises a digital counter; means responsive to saidfirst signals for presetting said counter, once during each ofsuccessive equal time periods AT, to hold a count state or number N,where N differs from a predetermined count state or number P by themagnitude of the number AX; means for supplying recurring pulses to saidcounter to cause it to count toward said predetermined count state P atall times that it does not hold the number P; means for producing asquarewave voltage which has one level when said counter is in its Pstate and a second level when said counter is not in its P state,whereby the duty cycle and the average value of said squarewave voltagevaries in proportion to changes in the number AX; and means forconverting said squarewave voltage into a dc. voltage which variesaccording to changes in said average value; the periods AT being shorterthan said periods AT and the period of said recurring pulses beingshorter than said periods A T.

5. In combination with means for producing a first set of signals XSCdigitally representing the commanded position of a member movable alongan axis, means for producing a second set of signals digitallyrepresenting a changeable number AX, means responsive to said second setof signals for changing said first signals to increment the number XSCby the amount AX during each of successive equal time periods AT, aD-to-A converter for converting said first signals into acorrespondingly changing analog command signal, and servo drive meansresponsive to said command signal and including position feedback meansfor driving said member along the axis to keep its actual positionsubstantially in agreement with that represented by the XSC number, theimprovement which comprises:

a. means responsive to said second set of signals for producing a feedforward analog signal which dynamically varies to be proportional tosaid number AX as the latter changes, and

b. means for additively combining said feed forward signal with saidcommand signal and making said servo drive means responsive to theeffective sum thereof.

6. The combination set forth in claim further characterized in that saidservo drive means has a transfer function generally representable by A/swhere A represents gain and s is a derivative operator, and said means(a) includes means for making the factor of proportionality relatingsaid feed forward signal to the number AX substantially equal to l/A,such that the feed forward signal in magnitude and polarity isapproximately equal to AX/A.

7. in a system having a servo drive with feedback to move a member alongan axis, and means for supplying a primary input signal to said servodrive which changes during each of successive equal periods AT an amountcorresponding to a changeable number AX represented by a set of digitalsignals, the combination comprising a. means responsive to said digitalsignals for producing an analog signal substantially instantaneouslyproportional to the number AX, and

b. means for supplying said analog signal as a secondary input to saidservo drive and in a sense to additively supplement said primary inputsignal.

8. In a positioning system having a servo drive with feedback to move amember along an axis, in response to a primary input signal; togetherwith means for changing said primary input signal, during each ofsuccessive equal time periods AT, an amount corresponding to achangeable number AX represented by a set of digital signals; thecombination comprising a. a digital counter, b. means responsive to saiddigital signals to preset said counter to a count state N which differsfrom a predetermined count state P by the amount AX,

c. a source of regularly recurring pulses,

d. means responsive to output signals from said counter and representingits count state for supplying said pulses to said counter to make thelatter count toward said count state P at all times when it is not instate P,

e. means for producing a squarewave signal having first and secondlevels when said counter respectively is and is not in said count stateP,

f. means for converting said squarewave signal into an analog signalwhich varies according to the duty cycle of said squarewave, and

g. means for supplying said analog signal as a supplementary inputsignal to said servo drive and causing the latter to respond to theadditive combination of the primary and supplementary input signals.

9. The combination set forth in claim 8 further char acterized in thatsaid means (d) and (e) are constituted by a decoder connected to receiveoutput signals from the counter which represent its count state andincluding means for producing a response signal having first or secondlevels when the counter respectively is or is not in said count state P,said response signalbeing said squarewave, and a gate having its outputconnected to the counting input of said counter and having two inputsrespectively connected to receive (i) said recurring pulses and (ii)said response signal.

10. The combination set forth in claim 8 further characterized in thatsaid number AX may be positive or negative and its sign represented by 0or 1 level of one bit signal in said set of digital signals, and saidmeans (2) includes el. means for making said squarewave signal have afirst positive level when said one bit signal is a 1 apdsa sls qnts isqtitii cq s a 22. means for making said squarewave signal have a secondnegative level, equal in magnitude to the first, when said one bitsignal is a 0 and said counter is not in its count state P, and H 7 e3.means for making said squarewave signal have a third, zero level whensaid counter is in its count state P.

l 1. The combination set forth in claim 8 further characterized in thatsaid predetermined count state P is zero.

12. The combination set forth in claim 11 further characterized in thatsaid presetting means sets said counter to a count state N which isequal to the number AX, and said counter is a down counter which countsdownwardly in response to said recurring pulses.

13. The combination set forth in claim 11 further characterized in thatsaid presetting means sets said counter to a count state N which is thecomplement of the number AX, and said counter is an upward counter whichcounts upwardly in response to said recurring pulses.

14. The method of moving a member along an axis to keep its positiondynamically in agreement with a changing position command, said methodcomprising the steps of changes in the number XSC,

f. converting said second set of signals into an analog feed forwardsignal which varies according to the changes in the number AX, and

g. utilizing said command and feed forward signals as additive inputs toa servo positioning drive means coupled to move the member along theaxis, said drive means having a position feedback loop, whereby thefollowing error between the actual position and the commanded positionrepresented by said first set of signals is reduced.

1. In a positioning system for moving an element to successive positionsalong an axis in accordance with numerical commands, the combinationcomprising a. means for measuring off successive equal time periodsDelta T, b. means for producing first signals digitally rePresenting achangeable number Delta X, where Delta X is proportional to the velocityVx at which said element is to be moved, c. means for producing secondsignals digitally representing a changeable command number XSC and forchanging it by the amount Delta X during each period Delta T, d. meansfor converting said second signals into a correspondingly changinganalog command signal, e. means for producing an analog feedback signalrepresenting the actual position of the element along the axis, f. meansresponsive to said first signals for producing an analog feed forwardsignal proportional to the changeable number Delta X, g. means forcombining said feedback signal subtractively and said feed forwardsignal additively with said command signal to produce a modified errorsignal, and h. amplifier and motor means responsive to said error signalfor driving said element along the path.
 2. The combination set forth inclaim 1 further including means for adjusting the constant ofproportionality relating said feed forward signal to the number Delta Xrepresented by said first signals.
 3. The combination set forth in claim1 further characterized in that said adjusting means is adjusted to makesaid constant of proportionality substantially equal to 1/A, where A isthe gain in the generalized transfer function A/s of said amplifier andmotor means.
 4. The combination set forth in claim 1 furthercharacterized in that said means (f) comprises a digital counter; meansresponsive to said first signals for presetting said counter, onceduring each of successive equal time periods Delta T'', to hold a countstate or number N, where N differs from a predetermined count state ornumber P by the magnitude of the number Delta X; means for supplyingrecurring pulses to said counter to cause it to count toward saidpredetermined count state P at all times that it does not hold thenumber P; means for producing a squarewave voltage which has one levelwhen said counter is in its P state and a second level when said counteris not in its P state, whereby the duty cycle and the average value ofsaid squarewave voltage varies in proportion to changes in the numberDelta X; and means for converting said squarewave voltage into a dc.voltage which varies according to changes in said average value; theperiods Delta T'' being shorter than said periods Delta T and the periodof said recurring pulses being shorter than said periods Delta T''. 5.In combination with means for producing a first set of signals XSCdigitally representing the commanded position of a member movable alongan axis, means for producing a second set of signals digitallyrepresenting a changeable number Delta X, means responsive to saidsecond set of signals for changing said first signals to increment thenumber XSC by the amount Delta X during each of successive equal timeperiods Delta T, a D-to-A converter for converting said first signalsinto a correspondingly changing analog command signal, and servo drivemeans responsive to said command signal and including position feedbackmeans for driving said member along the axis to keep its actual positionsubstantially in agreement with that represented by the XSC number, theimprovement which comprises: a. means responsive to said second set ofsignals for producing a feed forward analog signal which dynamicallyvaries to be proportional to said number Delta X as the latter changes,and b. means for additively combining said feed forward signal with saidcommand signal and making said servo drive means responsive to theeffective sum thereof.
 6. The combination set forth in claim 5 furthercharacterized in that said servo drive means has a transfer functiongenerally representable by A/s where A represents gain and s is aderivative operator, and said means (a) includes means for making thefactor of proportionality relating said feed forward signal To thenumber Delta X substantially equal to 1/A, such that the feed forwardsignal in magnitude and polarity is approximately equal to Delta X/A. 7.In a system having a servo drive with feedback to move a member along anaxis, and means for supplying a primary input signal to said servo drivewhich changes during each of successive equal periods Delta T an amountcorresponding to a changeable number Delta X represented by a set ofdigital signals, the combination comprising a. means responsive to saiddigital signals for producing an analog signal substantiallyinstantaneously proportional to the number Delta X, and b. means forsupplying said analog signal as a secondary input to said servo driveand in a sense to additively supplement said primary input signal.
 8. Ina positioning system having a servo drive with feedback to move a memberalong an axis, in response to a primary input signal; together withmeans for changing said primary input signal, during each of successiveequal time periods Delta T, an amount corresponding to a changeablenumber Delta X represented by a set of digital signals; the combinationcomprising a. a digital counter, b. means responsive to said digitalsignals to preset said counter to a count state N which differs from apredetermined count state P by the amount Delta X, c. a source ofregularly recurring pulses, d. means responsive to output signals fromsaid counter and representing its count state for supplying said pulsesto said counter to make the latter count toward said count state P atall times when it is not in state P, e. means for producing a squarewavesignal having first and second levels when said counter respectively isand is not in said count state P, f. means for converting saidsquarewave signal into an analog signal which varies according to theduty cycle of said squarewave, and g. means for supplying said analogsignal as a supplementary input signal to said servo drive and causingthe latter to respond to the additive combination of the primary andsupplementary input signals.
 9. The combination set forth in claim 8further characterized in that said means (d) and (e) are constituted bya decoder connected to receive output signals from the counter whichrepresent its count state and including means for producing a responsesignal having first or second levels when the counter respectively is oris not in said count state P, said response signal being saidsquarewave, and a gate having its output connected to the counting inputof said counter and having two inputs respectively connected to receive(i) said recurring pulses and (ii) said response signal.
 10. Thecombination set forth in claim 8 further characterized in that saidnumber Delta X may be positive or negative and its sign represented by 0or 1 level of one bit signal in said set of digital signals, and saidmeans (e) includes e1. means for making said squarewave signal have afirst positive level when bit signal is a 1 and said counter is not inits count state P, e2. means for making said squarewave signal have asecond negative level, equal in magnitude to the first, when said bitsignal is a 0 and said counter is not in its count state P, and e3.means for making said squarewave signal have a third, zero level whensaid counter is in its count state P.
 11. The combination set forth inclaim 8 further characterized in that said predetermined count state Pis zero.
 12. The combination set forth in claim 11 further characterizedin that said presetting means sets said counter to a count state N whichis equal to the number Delta X, and said counter is a down counter whichcounts downwardly in response to said recurring pulses.
 13. Thecombination set forth in claim 11 further characterized in that saidpresetting means sets said counter to a count state N which is thecomplement of the number Delta X, and said counter Is an upward counterwhich counts upwardly in response to said recurring pulses.
 14. Themethod of moving a member along an axis to keep its position dynamicallyin agreement with a changing position command, said method comprisingthe steps of a. measuring off successive equal time periods Delta T, b.producing a first set of digital signals representing a commandedposition number XSC, c. producing a second set of digital signalsrepresenting a changeable increment number Delta X, where Delta X Vx/Delta T and Vx is the desired velocity of said member, d. utilizing saidsecond set of signals to change the first set of signals during eachperiod Delta T so as to change the number XSCi + Delta X XSCi 1, e.converting said first set of signals into an analog command signal whichvaries according to the changes in the number XSC, f. converting saidsecond set of signals into an analog feed forward signal which variesaccording to the changes in the number Delta X, and g. utilizing saidcommand and feed forward signals as additive inputs to a servopositioning drive means coupled to move the member along the axis, saiddrive means having a position feedback loop, whereby the following errorbetween the actual position and the commanded position represented bysaid first set of signals is reduced.